How much of a problem is white noise for the real-world usage of a DNN?
The white noise that fools DNNs isn't really white noise. It has been altered in the same way as the synthetic misclassified pictures have been altered. You have to change many input pixels in exactly such a way, that these little changes aren't perceptible, but propagated through the network add up to a misclassification. This is not going to happen by chance.